Hinton - The Fourth Dimension.pdf

THE EVIDENCES FOR A FOURTH DIMENSION 81
the strip of matter represented by CD and EF and a
multitude of rods like them can turn round the circular
circumference.
Thus this particular section of the sphere can turn
inside out, and what holds for any one section holds for
all. Hence in four dimensions the whole sphere can, if
extensible. turn inside out. Moreover, any part of it--
a bowl-shaped portion for instance—can turn inside out,
and so on round and round.
This is really no more than we had before in the
rotation about a plane, except that we see that the plane
can, in the case of an extensible matter, be curved, and still
play the part of an axis.
If we suppose the spherical shell to be of four-dimensional
matter, our representation will be a little different.
Let us suppose there to be a small thickness in the matter
in the fourth dimension. This would make no difference
in fig. 44, for that merely shows the view in the xyz
space. But when the x axis is let drop, and the w axis
comes in, then the rods CD and EF which represent the
matter of the shell, will have a certain thickness perpendicular
to the plane of the paper on which they are drawn.
If they have a thickness in the fourth dimension they will
show this thickness when looked at from the direction of
the w axis.
Supposing these rods, then, to be small slabs strung on
the circumference of the circle in fig. 45, we see that
there will not be in this case either any obstacle to their
turning round the circumference. We can have a shell
of extensible matter or of fluid material turning inside
out in four dimensions.
And we must remember than in four dimensions there
is no such thing as rotation round an axis. If we want to
investigate the motion of fluids in four dimensions we
must take a movement about an axis in our space, and